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In a life table, we consider the probability of a person dying from age x to x + 1, called qx. In the continuous case, we could also consider the conditional probability of a person who has attained age (x) dying between ages x and x + Δx, which is

where FX(x) is the cumulative distribution function of the continuous age-at-death random variable, X. As Δx tends to zero, so does this probability in the continuous case. The approximate force of mortality is this probability divided by Δx. If we let Δx tend to zero, we get the function for force of mortality, denoted by μ(x){\displaystyle \mu (x)}:

To understand conceptually how the force of mortality operates within a population, consider that the ages, x, where the probability density function fX(x) is zero, there is no chance of dying. Thus the force of mortality at these ages is zero. The force of mortality μ(x) uniquely defines a probability density function fX(x).

The force of mortality μ(x){\displaystyle \mu (x)} can be interpreted as the conditional density of failure at age x, while f(x) is the unconditional density of failure at age x.[1] The unconditional density of failure at age x is the product of the probability of survival to age x, and the conditional density of failure at age x, given survival to age x.

This is expressed in symbols as

μ(x)S(x)=fX(x){\displaystyle \,\mu (x)S(x)=f_{X}(x)}

or equivalently

μ(x)=fX(x)S(x).{\displaystyle \mu (x)={\frac {f_{X}(x)}{S(x)}}.}

In many instances, it is also desirable to determine the survival probability function when the force of mortality is known. To do this, integrate the force of mortality over the interval x to x + t